Question

A box of 1 L capacity is divided into two equal compartments by a thin partition which are filled with \(2 \mathrm{~g} \mathrm{H}_{2}\) and \(16 \mathrm{~g} \mathrm{CH}_{4}\) respectively. The pressure in each compartment is recorded as \(P \mathrm{~atm}\). The total pressure when partition is removed will be: (a) \(P\) (b) \(2 P\) (c) \(P / 2\) (d) \(P / 4\)

Short answer

The total pressure when the partition is removed will be P atm. Therefore, the correct answer is (a) P.

Step-by-step solution

Understand the Concept of Partial Pressures

In a mixture of non-reactive gases, each gas exerts pressure independently without being affected by the presence of the other gases. This concept is known as Dalton's Law of Partial Pressures. When the partition between the gases is removed, the total pressure of the mixture is the sum of the pressures that each gas would exert if it occupied the entire volume on its own. Here, since the volume doubles for each gas but the amount of gas doesn't change when the partition is removed, the total pressure will be the sum of the two pressures, each of which is now half of what it was before due to the doubling of volume for each gas.

Calculate the Final Pressure for Hydrogen (H2)

According to Boyle's Law, for a fixed amount of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. The hydrogen gas was originally at pressure P in half the volume. When the partition is removed, the volume available to the hydrogen doubles, so its pressure is halved. Hence, the new pressure exerted by hydrogen is P/2 atm.

Calculate the Final Pressure for Methane (CH4)

Similarly, the methane gas was at pressure P in its half of the volume. When the partition is removed, its new pressure becomes P/2 atm for the same reason explained in the previous step.

Calculate the Total Pressure

Now, we can calculate the total pressure in the box by adding the individual pressures exerted by hydrogen and methane after the partition is removed. This gives us (P/2) + (P/2) = P atm as the total pressure in the box.

Key concepts

Partial Pressures

Understanding partial pressures is essential when dealing with gaseous mixtures. Each gas in a blend behaves as though it occupies the space alone and exerts its own pressure, known as its partial pressure.

This principle is a consequence of Dalton's Law of Partial Pressures, which states that in a mixture of non-reactive gases, the total pressure is the sum of the partial pressures of the individual gases. This law is fundamental for solving problems related to gaseous behavior in sealed environments, as the individual pressures contribute to the overall pressure without influencing each other.

When the partial pressures are understood, it becomes easier to predict the behavior of a gas mixture when conditions change, such as the removal of a partition in a box containing different gases.

Boyle's Law

Boyle's Law is a fundamental concept within the study of gases, often explored in Physical Chemistry. It states that for a given mass of an ideal gas, at constant temperature, the volume of the gas is inversely proportional to its pressure.The formula representing Boyle's Law is \( P_1V_1 = P_2V_2 \), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume, respectively. This means if you increase the volume of the gas, the pressure will decrease, provided temperature remains constant. Boyle's Law is useful when predicting gas behavior upon changing the volume, as seen with the removal of a partition in a compartmentalized container.

Gaseous Mixtures

Gaseous mixtures, such as the combination of hydrogen and methane in our exercise, exhibit properties that are the summation of their individual components' properties. In these mixtures, each gas follows the individual gas laws, like Boyle's Law and Dalton's Law of Partial Pressures, independently.

For students and professionals tackling scenarios involving gaseous mixtures, it is crucial to have a grasp on how to calculate properties like total pressure, volume, and temperature. The behavior of these mixtures applies to a variety of real-world situations, from industrial processes to environmental science.

Physical Chemistry JEE

Students preparing for competitive exams like the Joint Entrance Examination (JEE) must gain a strong understanding of concepts like Dalton's Law and Boyle's Law. Physical Chemistry encompasses thermodynamics, equilibrium, kinetics, and the behavior of gases, all of which are pivotal topics for the JEE.

These principles do not stand-alone; they interrelate and form the basis for more complex applications in chemistry. Solving problems such as the one about the gas mixture in the divided box helps students develop the analytical skills required to succeed in exams like the JEE and in future scientific endeavors.